There is also magic taking place. In amplitude modulation, the thing to remember is that the actual power is going up and down as the signal gets louder and softer, so silence means hardly any modulation, and the loudest signal maxes out the transmitter. As radio at the listener end is the wanted signal mixed in with all the interference - as in man made, and natural, like lightning - you don't want the wanted signal disappearing into the noise, so broadcast stations use compressors to bring up the quiet bits to near the volume of the loud bits. This makes speech appear more round and punchy - the sound the US gave to the world as 'radio'. AM is of course quite inefficient. The power is the sort of oomph, and the content - speech or music is the fine detail.
A strange effects also occurs - while you have a centre spot frequency, with the 'power' - you also get two smaller bands of energy containing the fine detail. I'm clearly oversimplifying this, by the way - but good for some research if you want). These two bands either side of the carrier carry the detail, the centre sort of operates as the railway track - and doesn't really do much we need at the receiver. Somebody had the bright idea you could remove the centre bit, the 'carrier' wave, and then recreate it at ever radio listening. It meant that the power wasted in the centre could be put into the side bands, making it more efficient. That became double side, band, suppressed carrier. DSB. They then tried removing one of the side bands too - and this became the standard for short wave communications. USB -upper side band, or lower side band USB.
The downside is a reduction in the audio bandwidth - think of it like an orchestra recording - AM has every instrument from the double basses to the piccolos. Single Side Band, with suppressed carrier doesn't have any double basses, only the high strings of the cellos, and the piccolo players vanish as do most of the flutes.
Worse though is that if you remove (for example) a carrier wave on 14.3001MHz, then the receiver has to recreate that 14.301MHz signal and mix it with the incoming sidebands to magically recreate the original. Trouble is, most receivers are not actually that accurate, or at least weren't, so the mixing went wrong and people's voices would become Micky Mouse, or turn a soprano into a bass baritone. They cured this with a knob marked RIT (receiver incremental tuning) which allows the listener to adjust the fine tuning and restore a voice more realistic to the one the other end.
So you need a carrier wave to 'launch' a signal containing the important data, speech or music, or tones and stuff, but as that process creates these phantom extra signals, you can just cut them out and use 100% of your power on the vital bit. So it's magic. You need something, but can remove it. That's pretty amazing, and was Marconi's Star Trek transporter.
This is the sort of science that bends your brain until suddenly, it doesn't. All my HF transmitters have been single side band since the first in 1980. AM is broadcast radio, like Voice of America and the BBC World Service. Yesterday I got a Xeigu HF radio and it has AM too. I tried it with my old HF Icom - and I transmitted AM for the first time in 44 years. I was amazed how close it was to FM. I should not have been surprised, because aircraft never made the change to FM like most other radio systems.
To be precise, we talk about modulation always linked to the mode - so that's the M bit. Amplitude Modulation, up and down in power and Frequency Modulation, up and down in frequency, but constant power. FM, looked at on a scope is a solid spike, and the width of the spike varies with content. AM spikes go higher and lower. AM also has one big spike, and two smaller spikes, spaced from the centre one - these are the side bands. You pick one, and reduce the other two and you have the SSB we use all the time. To wind you up even more, 14MHz and below, hams use the lower one, and above, they use the higher one as a sort of convention, but not I have noticed anymore, a rule.
We actually do have mixing in radio. It exists in most radio designs. Take one frequency, add another and the two mix in bizarre ways. The two add together in frequency terms and create a new one - the sum of the two, but they also create another lower one - the difference of the two. In practice if you are trying to create a new one, the other is a nuisance, so you then filter that one out.
In many practical applications mixing is bad! I work with radio microphones and many users with two or three channels have no issues. However they might hire in extra six they need ten, and you cannot just randomly pick frequencies because all the mixing of each transmitter mixes with the others and all sorts of phantom signals appear, often blotting out another, but changing as people move around. People like Shure and Sennheiser products lists of frequencies that can be used together without all these sums and differences causing issues. What you cannot do is remove a faulty Sennheiser and replace it with a Sony, Shure, Lectrosonic or other one on the same frequency - their receivers will work differently and mess it all up!
You were quite correct. Modulation is the mixing of one signal with another. We just steer clear of using mixing in this way, because we use 'mixing' differently. Confused? I bet!